Ian White MRC Biostatistics Unit, Cambridge An overview of meta-analysis in Stata Part II: multivariate meta-analysis Ian White MRC Biostatistics Unit, Cambridge Stata Users’ Group London, 10th September 2010 Abstract: An overview of meta-analysis in Stata A comprehensive range of user-written commands for meta-analysis is available in Stata, and documented in detail in the recent book Meta-Analysis in Stata (Stata Press 2009).The purpose of this session will be to describe these commands, with a focus on recent developments and areas in which further work is needed. We will define systematic reviews and meta-analyses, and introduce the metan command, which is the main Stata meta-analysis command. We will distinguish between meta-analyses of randomized controlled trials and observational studies, and discuss the additional complexities inherent in systematic reviews of the latter. Meta-analyses are often complicated by heterogeneity – variation between the results of different studies beyond that expected due to sampling variation alone. Meta-regression – implemented in the metareg command – can be used to explore reasons for heterogeneity, although its utility in medical research is limited by the modest numbers of studies typically included in meta-analyses, and the many possible reasons for heterogeneity. Heterogeneity is a striking feature of meta-analyses of diagnostic test accuracy studies. We will describe how to use the midas and metandi command to display and meta-analyse the results of such studies. Many meta-analysis problems involve combining estimates of more than one quantity: for example, treatment effects on different outcomes, or contrasts between more than two groups. Such problems can be tackled using multivariate meta-analysis, implemented in the mvmeta command. We will describe how the model is fitted, when it may be superior to a set of univariate meta-analyses, and illustrate its application in a variety of settings.

Plan Example 1: Berkey data Multivariate random-effects meta-analysis model Situations where it could be used Software: mvmeta A problem: unknown within-study correlation Example 2: fibrinogen software: mvmeta_make Multivariate vs. univariate

Example from Berkey et al (1998) 5 trials comparing a surgical with a non-surgical procedure for treating periodontal disease 2 outcomes: “probing depth” (PD) “attachment level” (AL) trial y1 s1 y2 s2 corr 1 0.47 0.09 -0.32 0.39 2 0.20 0.08 -0.60 0.03 0.42 3 0.40 0.05 -0.12 0.04 0.41 4 0.26 -0.31 0.43 5 0.56 0.12 -0.39 0.17 0.34 y1,y2 - treatment effects for PD, AL; s1,s2 - standard errors

Berkey data (1) Could analyse the outcomes one by one Random effects weight Random effects weight Study ID Study ID 1 17.82 17.82 1 19.71 19.71 2 19.84 19.84 2 22.05 22.05 3 25.64 25.64 3 21.83 21.83 4 24.08 24.08 4 21.79 21.79 5 12.62 12.62 5 14.61 14.61 Overall 100.00 100.00 Overall 100.00 100.00 I2 = 68.8%, p = 0.012 I2 = 96.4%, p = 0.000 .5 1 -1 -.5 Mean improvement in probing depth Mean improvement in attachment level

Berkey data (2) Dots mark the point estimates for the 5 studies -.8 -.6 -.4 -.2 Mean improvement in attachment level .2 .4 .6 .8 Mean improvement in probing depth Dots mark the point estimates for the 5 studies Bubbles show 50% confidence regions Note the positive within-study correlation (0.3-0.4 for all studies) bubble.ado, available on my website 3 4 5 1 2

One or two stages? I’m assuming a two-stage meta-analysis (as in the Berkey data): 1st stage: compute results for each study 2nd stage: use these results as “data” makes a Normal approximation to the within-study log-likelihoods One-stage meta-analysis is possible if we have individual participant data (IPD), but can be computationally horrible (Smith et al 2005) we’ll use the two-stage method even with IPD

Bivariate meta-analysis: data Data from ith study: yi1, yi2 – estimates for 1st, 2nd outcomes si1, si2 – their standard errors but we also need the correlation rWi of yi1 and yi2 It’s often most convenient to use matrix notation: estimate with within-study variance NB yi1 or yi2 can be missing.

Bivariate meta-analysis: the model Data from ith study: yi – vector of estimates Si – variance-covariance matrix Model is yi ~ N(m, Si+S) Total variance = within + between variance: known to be estimated

Bivariate meta-analysis: 2 correlations Within-study correlation rWi one per study should be known from 1st stage of meta-analysis but often unknown: discussed later Between-study correlation rB overall parameter to be estimated

Multivariate meta-analysis: the model Data from ith study: yi – vector of estimates (p-dimensional) Si – variance-covariance matrix (pxp) Model is again yi ~ N(m, Si+S) Can also extend to meta-regression: e.g. yi ~ N(bxi, Si+S) xi is a q–dimensional vector of explanatory variables b is a pxq matrix containing the regression coefficients for each of the p outcomes more generally, can allow different x’s for different outcomes

When could multivariate meta-analysis be used? (1) Original applications: meta-analysis of randomised controlled trials (RCTs) several outcomes of interest some trials report more than one outcome “data” are treatment effects on each outcome in each study (some may be missing) data are correlated within studies because outcomes are correlated also used in health economics for cost and effect (Pinto et al, 2005)

When could multivariate meta-analysis be used? (2) Meta-analysis of diagnostic accuracy studies “data” are sensitivity and specificity in each study data are uncorrelated within studies because they refer to different subgroups still likely to be correlated between studies See Roger’s talk sparse data often invalidates Normal approximation best to use metandi

When could multivariate meta-analysis be used? (3) Meta-analysis of RCTs comparing more than two treatments “data” are treatment effects for each treatment compared to same control data are correlated within studies because they use same control group Similarly multiple treatments meta-analysis my current area of research

When could multivariate meta-analysis be used? (4) Meta-analysis of observational studies exploring shape of exposure-disease relationship if exposure is categorised, “data” could be contrasts between categories if fractional polynomial model is used, “data” would be coefficients of different model terms

Stata software for multivariate random-effects meta-analysis Can almost use xtmixed but you need to constrain the level 1 (co)variances not possible in xtmixed So I wrote mvmeta (White, 2009)

My program: mvmeta Analyses a data set containing point estimates with their (within-study) variances and covariances Utility mvmeta_make creates a data set in the correct format (demo later) Fits random-effects model uses ml to maximise the (restricted) likelihood using numerical derivatives between-studies variance-covariance matrix is parameterised via its Cholesky decomposition CIs are based on Normal distribution also offers method of moments estimation (Jackson et al, 2009)

Data format for mvmeta: Berkey data trial y1 y2 V11 V22 V12 1 0.47 -0.32 0.0075 0.0077 0.003 2 0.2 -0.6 0.0057 0.0008 0.0009 3 0.4 -0.12 0.0021 0.0014 0.0007 4 0.26 -0.31 0.0029 0.0015 5 0.56 -0.39 0.0148 0.0304 0.0072 y1, y2 treatment effects for PD, AL V11, V22 squared standard errors (si12, si22) V12 covariance (rWisi1si2)

Running mvmeta: Berkey data . mvmeta y V Note: using method reml Note: using variables y1 y2 Note: 5 observations on 2 variables [5 iterations] Number of obs = 5 Wald chi2(2) = 93.15 Log likelihood = 2.0823296 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Overall_mean | y1 | .3534282 .061272 5.77 0.000 .2333372 .4735191 y2 | -.3392152 .08927 -3.80 0.000 -.5141811 -.1642493 Estimated between-studies SDs and correlation matrix: SD y1 y2 y1 .1083191 1 .60879876 y2 .1806968 .60879876 1

Running mvmeta: method of moments . mvmeta y V, mm Note: using method mm (truncated) Note: using variables y1 y2 Note: 5 observations on 2 variables Multivariate meta-analysis Method = mm Number of dimensions = 2 Number of observations = 5 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- y1 | .3478429 .0557943 6.23 0.000 .238488 .4571978 y2 | -.3404843 .1131496 -3.01 0.003 -.5622534 -.1187152 Estimated between-studies SDs and correlation matrix: SD y1 y2 y1 .10102601 1 .74742532 y2 .23937024 .74742532 1

Running mvmeta: I2 I2 measures the impact of heterogeneity (Higgins & Thompson, 2002) . mvmeta1 y V, i2 [output omitted] I-squared statistics: -------------------------------------------------- Variable I-squared [95% Conf. Interval] y1 72% -45% 94% y2 94% 76% 98% (computed from estimated between and typical within variances) Requires updated mvmeta1

Running mvmeta: meta-regression . mvmeta1 y V publication_year, reml dof(n-2) Note: using method reml Note: using variables y1 y2 Note: 5 observations on 2 variables Variance-covariance matrix: unstructured [4 iterations] Multivariate meta-analysis Method = reml Number of dimensions = 2 Restricted log likelihood = -5.3778317 Number of observations = 5 Degrees of freedom = 3 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- y1 | publicatio~r | .0048615 .0222347 0.22 0.841 -.0658992 .0756221 _cons | .3587569 .0740749 4.84 0.017 .1230175 .5944963 y2 | publicatio~r | -.0115367 .0303001 -0.38 0.729 -.107965 .0848917 _cons | -.3357368 .0985988 -3.41 0.042 -.6495222 -.0219513

mvmeta: programming Basic parameters: Cholesky decomposition of the between-studies variance S Eliminate fixed parameters from (restricted) likelihood Maximise using ml, method d0 (can’t use lf for REML) Likelihood now coded in Mata Stata creates matrices yi , Si for each study & sends them to Mata

Estimating the within-study correlation ρwi Sometimes known to be 0 e.g. in diagnostic test studies where sens and spec are estimated on different subgroups Estimation usually requires IPD even then, not always trivial: e.g. for 2 outcomes in RCTs, can fit seemingly unrelated regressions, or observe ρwi = correlation of the outcomes Published literature never (?) reports ρwi not the objective of the original study difficult to estimate from summary data What do we do in a published literature meta-analysis if ρwi values are missing? 23

Unknown ρwi: possible solutions Ignore within-study correlation (set ρwi = 0) not advisable (Riley, 2009) Sensitivity analysis using a range of values can be time-consuming & confusing Use external evidence (e.g. IPD on one study) Bayesian approach (Nam et al., 2004) e.g. ρwi ~U(0,1) Some special cases where it can be done % survival at multiple time-points nested binary outcomes? Use an alternative model that models the ‘overall’ correlation (Riley et al., 2008) 24 24

Alternative bivariate model Standard model with overall rB and one rWi per study: Alternative model with one ‘overall’ correlation r: mvmeta1: corr(riley) option 25

Example: Fibrinogen Fibrinogen Studies Collaboration (2005) assembled IPD from 31 observational studies 154211 participants to explore the association between fibrinogen levels (measured in blood) and coronary heart disease We focus on exploring the shape of the association using grouped fibrinogen Data (IPD): Variable fg contains fibrinogen in 5 groups Studies are identified by variable cohort Time to CHD has been stset In each cohort, I want to run the Cox model xi: stcox age i.fg, strata(sex tr)

1st stage of meta-analysis: mvmeta_make Getting IPD into the right format can be the hardest bit I wrote mvmeta_make to do this It assumes the 1st stage of meta-analysis involves fitting a regression model

Fibrinogen data: using mvmeta_make Stata command within each study: xi: stcox age i.fg, strata(sex tr) Create meta-analysis data set: xi: mvmeta_make stcox age i.fg, strata(sex tr) by(cohort) usevars(i.fg) name(b V) saving(FSC2) Creates file FSC2.dta containing coefficients: b_Ifg_2, b_Ifg_3, b_Ifg_4, b_Ifg_5 variances and covariances: V_Ifg_2_Ifg_2, V_Ifg_2_Ifg_3 etc. We then run mvmeta b V on file FSC2.dta.

A problem: perfect prediction . tab fg allchd if cohort=="KORA_S3" Fibrinogen | Any CHD event? groups | 0 1 | Total -----------+----------------------+---------- 1 | 546 0 | 546 2 | 697 3 | 700 3 | 715 2 | 717 4 | 677 4 | 681 5 | 482 8 | 490 Total | 3,117 17 | 3,134 No events in the reference category Fit Cox model: HR for 2 vs 1 is 21.36 (se 0.91) – wrong

mvmeta_make: handling perfect prediction Recall: no events in fg=1 (reference) group stcox’s “fix” can yield large hazard ratios with small standard errors – and disaster for mvmeta! mvmeta_make implements a different “fix” in any study with perfect prediction: add a few observations, with very small weight, that “break” the perfect prediction all contrasts with fg=1 are large with large s.e. all other contrasts (e.g. fg=3 vs. fg=2) are correct Works fine for likelihood-based procedures (REML, ML, fixed-effect model) but not for method of moments

FSC: partial results of mvmeta_make . l c b* V_Ifg_2_Ifg_2 V_Ifg_3_Ifg_3 , clean noo cohort b_Ifg_2 b_Ifg_3 b_Ifg_4 b_Ifg_5 V_Ifg_~2 ~3_Ifg_3 ARIC 0.252 0.532 0.946 1.401 0.036 0.033 BRUN -0.184 -0.032 0.119 0.567 0.348 0.344 CAER 0.001 -0.529 -0.339 0.416 0.375 0.323 CHS 0.066 0.184 0.407 0.645 0.058 0.053 COPEN 0.078 0.406 0.544 1.088 0.101 0.083 EAS -0.113 0.456 0.456 0.875 0.065 0.054 FINRISKI -2.149 -0.264 -0.494 0.169 1.336 0.421 FRAM -0.039 0.170 0.420 1.053 0.042 0.038 GOTO 0.443 0.595 0.922 0.797 0.202 0.175 GOTO33 0.356 1.312 0.628 2.133 1.500 1.170 GRIPS 1.297 1.052 1.421 1.752 0.559 0.542 HONOL 0.323 0.545 0.681 0.540 0.132 0.122 KIHD -0.042 0.509 0.560 0.998 0.088 0.072 KORA_S2 -2.667 -2.524 -2.010 -1.767 1.337 0.584 KORA_S3 5.946 5.420 6.088 7.057 189.088 189.271 MALMO 0.123 0.371 0.506 0.936 0.071 0.058 ... Study with no events in fg=1 group: “perfect prediction”

FSC: results of mvmeta . mvmeta b V Number of obs = 31 Wald chi2(4) = 142.62 Log likelihood = -79.129029 Prob > chi2 = 0.0000 -------------------------------------------------------------------- | Coef. Std. Err. z P>|z| [95% Conf. Int.] -------------+------------------------------------------------------ Overall_mean | b_Ifg_2 | .1646353 .0787025 2.09 0.036 .0103813 .3188894 b_Ifg_3 | .3905063 .088062 4.43 0.000 .2179080 .5631047 b_Ifg_4 | .5612908 .0904966 6.20 0.000 .3839206 .7386609 b_Ifg_5 | .8998468 .0932989 9.64 0.000 .7169843 1.082709 Estimated between-studies variance matrix Sigma: b_Ifg_2 b_Ifg_3 b_Ifg_4 b_Ifg_5 b_Ifg_2 .04945818 b_Ifg_3 .06355581 .0836853 b_Ifg_4 .06689067 .08920553 .09570788 b_Ifg_5 .0506146 .07530983 .08501967 .1041611

FSC: graphical results Other choices of reference category give the same results.

Example 2: borrowing strength y2>0 ⇒ y1 missing y2<0 ⇒ y1 observed Pooling the observed y1 can’t be a good way to estimate m1 Bivariate model helps: assumes a linear regression of m1 on m2 assumes data are missing at random Bivariate model can avoid bias & increase precision (“Borrowing strength”) Study Log hazard ratio (mutant vs. normal p53 gene) Disease-free survival Overall survival y1 s1 y2 s2 1 -0.58 0.56 -0.18 2 0.79 0.24 3 0.21 0.66 4 -1.02 0.39 -0.63 0.29 5 1.01 0.48 6 -0.69 0.40 -0.64

Multivariate vs. univariate meta-analysis Advantages: “borrowing strength” avoiding bias from selective outcome reporting Joint confidence / prediction intervals Functions of estimates Longitudinal data Coherence Disadvantages: more computationally complex boundary solutions for rB unknown within-study correlations more assumptions

Getting mvmeta mvmeta is in the SJ Current update mvmeta1 is available on my website (includes meta-regression, I2, structured S, speed & other improvements) net from http://www.mrc‑bsu.cam.ac.uk/IW_Stata bubble is also available

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